Laplace equations and the Strong Lefschetz Property

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1 Laplace equations and the Strong Lefschetz Property Roberta Di Gennaro, Giovanna Ilardi, Jean Vallès To cite this version: Roberta Di Gennaro, Giovanna Ilardi, Jean Vallès. Laplace equations and the Strong Lefschetz Property <hal v1> HAL Id: hal Submitted on 8 Oct 2012 (v1), last revised 20 Jan 2014 (v4) HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

2 LAPLACE EQUATIONS AND THE STRONG LEFSCHETZ PROPERTY ROBERTA DI GENNARO, GIOVANNA ILARDI AND JEAN VALLÈS Abstract. As in [9] we explore the link between artinian ideals that fail the Weak Lefschetz Property (WLP) in degree d 1 and projections of the Veronese varieties satisfying a Laplace equation of order d 1. We extend this link to the more general situation of artinian ideals failing the Strong Lefschetz Property (SLP) at the range k 1 in degree d k. This generalization is not artificial; indeed for k = 2 it is related to the so-called Terao s conjecture about free arrangements. We reformulate the Terao s conjecture for line arrangements in terms of artinian ideals failing the SLP at the range 2. Using this new link we propose non toric examples of ideals failing the SLP at the range 2. Moreover we add a new characterization of WLP or SLP in terms of singular hypersurfaces (see thm. 3.1). Thanks to this characterization we produce many toric examples that fail the WLP and also the SLP at the range 2. October 8, Introduction The study of the surfaces satisfying Laplace equations was developed in the last century by Togliatti [16] and Terracini [15]. Togliatti [16] gave a complete classification of the rational surfaces embedded with linear systems of plane cubics and satisfying a Laplace equation of order two. For a linear system of cubic surfaces that satisfy a Laplace equation of degree two, a preliminary work was done by Ilardi in [7], and a complete classification of such smooth threefolds is achieved by Miro-Roig, Mezzetti and Ottaviani ( [9], Thm. 4.11). In order to do it they establish and study the link between artinian ideals that fail the weak Leftschetz property (WLP) in degree d 1 and the projections of Veronese varieties satisfying at least one Laplace equation of order d 1 2. In this paper we extend this link to the more general situation of artinian ideals failing the Strong Lefschetz Property (SLP) at the range k 1 in degree d k. To explain more precisely what we will do in this paper, let us interrupt briefly this introduction to fix some notations and say, at least, what mean WLP and SLP. Let R = k[x 0,, x n ] be the graded polynomial ring in n + 1 variables over a field of characteristic zero k. Let us denote by r t the dimension (that is ( ) n+t n ) of the vector space R t. m Let A = R/I = be a graded artinian algebra. Note that A is finite dimensional over k. Definition 1.1. Let l be a general linear form. The artinian algebra A (or the artinian ideal I) has the Weak Lefschetz Property (WLP) if the homomorphism induced by the Key words and phrases. SLP, WLP. Third author partially supported by ANR-09-JCJC INTERLOW, ANR GEOLMI and the project F.A.R.O. 2010: Algebre di Hopf, differenziali e di vertice in geometria, topologia e teorie di campo classiche e quantistiche. 1 i=0 A i

3 2 ROBERTA DI GENNARO, GIOVANNA ILARDI AND JEAN VALLÈS multiplication by l, l : A i A i+1, has maximal rank (i.e. is injective or surjective) for all i.the artinian algebra A (or the artinian ideal I) has the Strong Lefschetz Property (SLP) if, l k : A i A i+k, has maximal rank (i.e. is injective or surjective) for all i and k. The most interesting artinian ideals are certainly those that fail the WLP in some degree or the SLP at some range and degree. To be precise, Definition 1.2. The artinian algebra A (or the artinian ideal I) fails the WLP in degree i if for a general linear form l the multiplication map l : A i A i+1, has not maximal rank. The artinian algebra A (or the artinian ideal I) fails the SLP in degree i at the range k if for a general linear form l the multiplication map has not maximal rank. l k : A i A i+k, Remark. It is clear that the SLP at the rang k = 1 corresponds to the WLP. So, as we wrote above, we study the link between artinian ideals I = (F 1,, F r ) k[x 0,, x n ] generated by r (r r d r d k ) homogeneous forms of degree d that fail SLP in degree d k at the range k 1 and projections (from the subspace P(< F 1,..., F r >)) of the Veronese variety v d (P n ) P (n+d d ) r 1 satisfying at least one Laplace equation of order d k (i.e. such that the general tangent space of order d k, generated by partial derivatives of order d k, has projective dimension < r d k ). This generalization is meaningful in the topic of hyperplane arrangements and more precisely for line arrangements. Indeed, one associates canonically to a non degenerated line arrangement A = {l 1,, l r } in P 2 an artinian ideal I = (l1 d,, ld r) k[x, y, z]. The Terao s conjecture which affirms that the freeness of any arrangement depends only on its combinatorics (see for instance [12] for a good definition of combinatorics of an arrangement) can be reformulated in terms of failure of SLP at the range 2. This link allows us to give new examples of such artinian ideals. The main idea to relate the notions of WLP or SLP (at the range k) and the projections of Veronese (from a subspace P(< F 1,..., F r >)) satisfying a Laplace equation is to consider the following exact sequence: 0 K O r P n (F 1,,F r) O P n(d) 0, to tensor it by O L k for a general linear form L and to compute the dimension of the space H 0 (K O L k). In the particular case H 0 (O L k(d)) = r the failure of SLP at the range k corresponds to the existence of special singular hypersurfaces. Thanks to this link and by exploiting an idea given in [8] to explain why Togliatti s surface verifies a Laplace equation, we produce many examples of artinian ideals that fail the WLP (i.e. the SLP at the range 1) or the SLP at the range Link WLP, SLP and Veronese varieties First of all we extend the key lemma 2.3 in [9] to SLP.

4 LAPLACE EQUATIONS AND THE STRONG LEFSCHETZ PROPERTY 3 Proposition 2.1. Let I = (F 1,, F r ) R be an artinian ideal generated by r r d r d k (where k is a fixed integer greater than 1) homogeneous forms of degree d. Let L be a general linear form, let B = R/(L k ) and let J = (f 1,, f r ) where the f i is the image of F i in B. Then the homomorphism of multiplication by L k φ d k : (R/I) d k (R/I) d has not maximal rank if and only if f 1,, f r are linearly dependent. Remark. We point out that when r = r d r d k the map is injective if and only if it is surjective. Then in that case we will show later that failing SLP (in degree d k at the range k) is equivalent to the existence of a hypersurface of degree d with multiplicity d k + 1 at a general point in the orthogonal vector space of the vector subspace < F 1,, F r > R d (see Thm. 3.1). Proof. We note that (R/I) d i R d i when i < d. Consider the exact sequence 0 [I:Lk ] I R I L k R I (k) R (k) 0. (I,L k ) R The cokernel of φ d k is ( (I,L k ) ) d. According to the upper bound of r we have dim(r/i) d k dim(r/i) d. Hence φ d k is not of maximal rank if and only if φ d k is not injective, if and only if rk(φ d k ) < ( ) n+d k n, if and only if dim(r/(i, L k )) d = dimb d dimj d = ( ) n 1+d n 1 dimjd > dim(r/i) d ( ) ( n+d k n = n+d ) ( n r n+d k ) n. Therefore φ d k is not injective if and only if ( ) ( ) ( ) n + d n 1 + d n + d k r dimj d >. n n 1 n Since the right-hand side is equal to ( ) ( n 1+d d 1 n+d k ) d k which is 0 when k 1 the map φ d k is injective if and only if dim C < f 1,, f r >< r. According to this proposition we can reformulate the WLP and the SLP in the following form. Proposition 2.2. Let I = (F 1,, F r ) R be an artinian ideal generated by r r d r d k (where k 1 is a fixed integer) homogeneous forms of degree d and K the vector bundle defined by the exact sequence 0 K O r P n (F 1,,F r) O P n(d) 0. Then I fails the SLP in degree d k at the range k if for a general linear form L we have H 0 (K O L k) 0. Let us briefly explain now the link with projections of v d (P n ) (a more complete description is done in the next section). The ideal I = (F 1,, F r ) with deg(f i ) = d fails the SLP at the range k in degree d k when for a general linear form L it exists a 1,, a r complex numbers and a form G of degree d k such that a 1 F a r F r = L k G. In other terms it means that the projective d k tangent space to v d (P n ) at the point [L d ] intersects P(< F 1,, F r >) (since this space correponds to degree d forms divisible by L k ). Let us consider the projection map P(R d ) \ P(< F 1, F r >) π P(R d / < F 1, F r >). Let us denote by X n,(i 1 ) d the image of v d (P n ) by π, like in [9]. The dimension of the d k tangent space to X n,(i 1 ) d is strictly less than the dimension of the d k tangent spaces to v d (P n ) since this last one meet the center of projection. In that case,

5 4 ROBERTA DI GENNARO, GIOVANNA ILARDI AND JEAN VALLÈS X n,(i 1 ) d satisfies a Laplace equation, i.e. there is a linear relation between the partial derivatives of order d k. We propose an extended version of Tea theorem proved in [9]. Theorem 2.3. Let I = (F 1,, F r ) R be an artinian ideal generated by r homogeneous polynomials of degree d. Let 1 k < d a positive integer. If r r d r d k then the following conditions are equivalent: (1) The ideal I fails the SLP at the range k in degree d k. (2) The homogeneous forms F 1,, F r become linearly dependent on a general multiple hyperplane L k of P n (i.e. P(< F 1,, F r >) T d k [L d ] v d ). (3) The n-dimensional variety X n,(i 1 ) d satisfies a Laplace equation of order d k. Proof. We have already seen in the above proposition 2.1 that (1) is equivalent to (2). As explained before, the fact that the space P(< F 1,, F r >) meets the general d k tangent space of the Veronese v d (P n ) is equivalent to say that there is a relation a 1 F a r F r = L k G for L general, in other words (3) is equivalent to (2). The key example is of course the one coming from the Togliatti s result (see for instance [1], Example 3.1): the ideal I = (x 3, y 3, z 3, xyz) fails the WLP in degree 2. There are many ways to prove it. One of them comes from the polarity on the rational normal cubic curve. It leads to a generalisation that gives one of the few non toric examples. Proposition 2.4 ( [17], Thm. 3.1). Let n 1 be an integer and l 1,, l 2n+1 be non concurrent linear forms on P 2. Then the ideal (l1 2n+1,, l2n+1, l i ) fails the WLP in degree 2n. i=1,,2n+1 Indeed on the general line L the 2n + 2 forms of degree 2n + 1 become dependent thanks to the polarity on the rational normal curve of degree 2n + 1. We propose the following conjecture. For n = 1 it is again the Togliatti s result. Conjecture. Let l i be non concurrent linear forms on P 2 and f be a form of degree 2n + 1 on P 2. Then, (l1 2n+1,, l2n+1, f) fails the WLP in degree 2n if and only if modulo linear combination 2n+1 f (l1 2n+1,, l2n+1, l i ). i=1 3. Singular hypersurfaces, WLP and SLP It is well known that according to the duality between a linear form L and its kernel {L = 0}, that is a hyperplane, the tangent spaces to the Veronese varieties can be interpreted as singular hypersurfaces. More precisely a hyperplane containing the tangent space T L dv d (P n ) corresponds in the dual space P n to a hypersurface with degree d that is singular at the point {L }. More generally a hyperplane containing T d k v L d d (P n ) corresponds to a hypersurface with degree d and multiplicity (d k + 1) at the point {L } (see for instance [5]). In order to precise this correspondence let us introduce some formalism. Let R 1 be a complex vector space of linear form of dimension n + 1 such that H 0 O P n(1) = R 1. We consider the Veronese embedding v d : P(R 1 ) = P n P(R d ) = P(H 0 O P n(d)). The

6 LAPLACE EQUATIONS AND THE STRONG LEFSCHETZ PROPERTY 5 image v d (P n ) is called Veronese n-fold of order d. At the point [L d ] v d (P n ), where L is an hyperplane in P(R 1 ), the (k + 1)-osculating space, 1 k d 1 (defined by the partial derivatives of order k) is identified to P(R k ), i.e. to degree d forms possessing a factorization L d k.g where G is a degree k form. The set of hyperplanes containing the (k + 1) tangent space at the point [L d ] is the projective space P((H 0 (m k+1 L (d)) ) of degree d hypersurfaces with multiplicity (k + 1) at the point L. Then we have v d (P n ) (k+1) = L d v d (P n )P((H 0 (m k+1 L (d)) =: X d,k+1, where m L is the ideal sheaf of the point L. It is the set of degree d hypersurfaces that are (k + 1)-singular at one point. For instance the dual variety of v d (P 2 ) is the discriminant variety that parametrizes the singular degree d curves. The smooth model of this variety X d,k+1 is a projective bundle usually called bundle of principal parts. Indeed the derivation map, dual of the multiplication map k : R d R k R d k, f a 0 + +a r=k k f 0...Xan n X a, Xan n X a 0 induces the following homomorphism of vector bundles on P(R 1 ): 0 R k O P n( d + k) k R d O P n P d,k+1 0. The first map can be represented by the matrix k of partial derivatives of order k of a Rd -basis. The projective bundle P(P d,k+1) P n P(Rd ) is the incidence variety {(x, f) k f(x) = 0} and the fiber over a point x P n is identified with the projective space P(H 0 (m k+1 x (d)) ) (of hypersurfaces of degree d passing through x with multiplicity (k + 1)) where m x is the ideal sheaf of the point x. Its image in P(Rd ) by the second projection is X d,k+1. It is an irreducible variety birational to P(P n,k+1 ) Projections of Veronese and sections of Principal bundles. Let R d be a vector subspace of dimension less or equal to r d r d k. We consider the projection map: Let K be the kernel of the induced map: P(R d ) \ P( ) P( R d ). 0 K O P n O P n(d) 0. We introduce the multiple incidence variety [point/hyperplane]: F k = {( X i Xi )k = 0} p k P n q k P n Applying the functor p k q k to the exact sequence defining K, we obtain 0 q k p k K O P n P d,k+1 R 1 q k p kk 0.

7 6 ROBERTA DI GENNARO, GIOVANNA ILARDI AND JEAN VALLÈS The surjective homomorphism Hom(R d, k) Hom(, k) gives an embedding P(Hom(, k)) P(Hom(R d, k)). Let us consider the diagram O P n O P n 0 R d k O P n( k) R d O P n P d,k R d k O P n( k) R d O P n R 1 q k p kk 0. It is important to understand that supp(r 1 q k p k K) = {x H0 (m k+1 x (d)) 0}. We remark that induces an artinian ideal since the map given by it is surjective. Then we have the following result when dim( ) = r d r d k. Theorem 3.1. Let I R be an artinian ideal generated by r homogeneous polynomials F 1,, F r of degree d. Let k d a positive integer. If r = r d r d k then the following conditions are equivalent: (1) The ideal I fails the SLP at the range k in degree d k. (2) The homogeneous forms F 1,, F r become linearly dependent on a general multiple hyperplane L k of P n. (3) The n-dimensional variety X n,(i 1 ) d satisfies at least one Laplace equation of order d k. R (4) It exists f d <F 1,,F such that f r> H0 (m d k+1 L ) for a general linear form L R 1. Proof. In the case of equality h 0 (K O L k) = h 1 (K O L k) for any linear form L. Then H 0 (K O L k) 0 (see proposition 2.2) implies H 1 (K O L k) 0 and the non vanishing of this last vector space means that L belongs to the support of the sheaf R 1 q k p k K. Since supp(r 1 q k p k K) = {x < F 1,, F r > H 0 (m k+1 x (d)) 0}, we have done. In the next section we will give many examples of ideals failing the WLP or the SLP by producing ad-hoc singular hypersurfaces. Moreover we will give also an exhaustive list of monomial ideals in degree 3 in four variables and degree 4 in three variables that fail the WLP Monomial ideals coming from singular hypersurfaces. In their nice paper about osculating spaces of Veronese surfaces, Lanteri and Mallavibarena remark that the equation of the curve given by three concurrent lines depends only on six monomials instead of seven. More precisely let us consider a cubic with a triple point at (a, b, c) passing through (1, 0, 0), (0, 1, 0) and ((0, 0, 1). Its equation is (bz cy)(az cx)(ay bx) = 0 and it depends on the monomials x 2 y, xy 2, x 2 z, xz 2, y 2 z, yz 2. So there is a non zero R form in 3 that is triple at a general point. By this way they explain the <x 3,y 3,z 3,xyz> Togliatti surprising phenomena ( [8], Thm. 4.1, [6] and [3]). We apply this idea in our context. In the monomial case being artinian for the ideal I means that it contains the forms x d 0,, xd n. Let us consider the (n + 1) basis point (1, 0,, 0), (0, 1,, 0),, (0, 0,, 0, 1) and let us assume that the number of monomials in I is equal to r d r d k for k 1 a fixed integer. Let us denote by I the vector subspace of R d generated by the monomials generating I. Then, as we said above, the ideal I fails the SLP at the range k in degree d k if and only if there exists at any point x = (a 0,, a n ) a hypersurface of degree d with multiplicity d k + 1 at x given by a form in R d / I.

8 LAPLACE EQUATIONS AND THE STRONG LEFSCHETZ PROPERTY 7 Let us consider first the case k = 1, i.e. the WLP case. Then the wanted singular hypersurface at x is given by d hyperplanes meeting at x and passing through the n+1 basis points. If this equation depends only on the monomials of R d / I then the image of v d (P n ) by π : P(R d ) \ P( I ) P(R d / I ) verifies a Laplace equation. Let us give some examples. Case n = 2, d = 4, k = 1. We describe exhaustively the monomial ideals (x 4, y 4, z 4, f, g) k[x, y, z] of degree 4 that do not verify the WLP. Theorem 3.2. Up to permutation of variables the monomial ideals generated by 5 forms that fail the WLP in degree 3 are one of the following (x 4, y 4, z 4, x 3 z, x 3 y) and (x 4, y 4, z 4, x 2 y 2, xyz 2 ). Remark. Geometrically it is evident that the first ideal (x 4, y 4, z 4, x 3 z, x 3 y) fails the WLP. Indeed modulo a linear form L the restricted monomials x i ȳ j can be interpretated as points of the projective P 4 defined by the rational normal curve of degree four image of L by the Veronese map. Then the tangent line to the rational quartic curve (isomorphic to L) at the point [ x 4 ] contains the two points [ x 3 ȳ] and [ x 3 z]. This line meets the plane P(< x 4, ȳ 4, z 4 >) in one point; it implies that dim C < x 4, ȳ 4, z 4, x 3 ȳ, x 3 z > 4. It is not so evident for the second ideal to see that the line P(< x 2 ȳ 2, xȳ z 2 >) always (when we restrict at any line) meet the plane P(< x 4, ȳ 4, z 4 >). Proof. Let us consider the points (1, 0, 0), (0, 1, 0) and (0, 0, 1) and the degree 4 curves with a quartuple point in (a, b, c) passing through these three points. These curves are product of four lines: f(x, y, z) = (ay bx)(az cx)(cy bz)(α(ay bx) + β(az cx)). We develop f. We find the following coefficients for monomials. Figure 1. quartic with a quartuple point. f(x, y, z) = bc(βc 2 αab)x 2 yz + ac( αab β(ac + c 2 ))xy 2 z + ac 2 (2αb + βc)x 2 y 2 + bc 2 ( αb βc)x 3 y+a 2 b(2αb+βc)xyz 2 +a 2 b(2αb+βc)x 2 z 2 +b 2 c(αb+βc)x 3 z+αa 3 cy 3 z αa 2 c 2 xy 3 + a 2 ( αab + βc 2 )y 2 z 2 βa 2 bcyz 3 + βab 2 c.xz 3. Then, twelve monomials appear to write the wanted quartic with a quartuple point. The forms x 4, y 4, z 4 are missing. We want only ten forms. The coefficients of two forms have to vanish. The following possibilities appear:

9 8 ROBERTA DI GENNARO, GIOVANNA ILARDI AND JEAN VALLÈS α = 0, the remaining linear system is (x 4, y 4, z 4, y 3 z, xy 3 ). β = 0 the remaining linear system is (x 4, y 4, z 4, yz 3, xz 3 ). α 0 and β 0 but αb + βc = 0, the remaining linear system is (x 4, y 4, z 4, x 3 z, x 3 y). α 0 and β 0 but βc 2 = αab the remaining linear system is (x 4, y 4, z 4, x 2 yz, y 2 z 2 ). α 0 and β 0 but 2αb + βc = 0, the remaining linear system is (x 4, y 4, z 4, x 2 y 2, xyz 2 ). α 0 and β 0 but αab + β(ac + c 2 ) = 0, the remaining linear system is (x 4, y 4, z 4, x 2 z 2, xy 2 z). Case n = 2, d = 5, k = 1. We cannot apply the same technique to describe exhaustively the monomial ideals (x 5, y 5, z 5, f, g, h) k[x, y, z] of degree 5 that do not verify the WLP. Indeed the computations become too tricky. We can give some cases by geometric arguments. Proposition 3.3. The following monomial ideals (x 5, y 5, z 5, x 3 y 2, x 3 z 2, x 3 yz) and (x 5, y 5, z 5, x 4 z, x 4 y, m), where m is any monomial, fail the WLP in degree 4. Proof. Modulo a linear form L the restricted monomials x i ȳ j can be interpretated as points of the projective P 5 defined by the rational normal curve of degree 5 image of L by the Veronese map. Then the tangent line to the rational quintic curve (isomorphic to L) at the point [ x 5 ] contains the two points [ x 4 ȳ] and [ x 4 z]. This line meet the plane P(< x 5, ȳ 5, z 5 >) in one point; it implies that dim C < x 5, ȳ 5, z 5, x 4 ȳ, x 4 z, m > 5. In the same way the osculating plane at [ x 5 ] i.e. P(< x 3 ȳ 2, x 3 z 2, x 3 ȳ z >) meets the plane P(< x 5, ȳ 5, z 5 >) in one point. Case n = 3, d = 3, k = 1. We describe exhaustively the monomial ideals of degree 3 that do not verify the WLP. (x 3, y 3, z 3, t 3, f 1, f 2, f 3, f 4, f 5, f 6 ) k[x, y, z, t] Theorem 3.4. Up to permutation of variables the monomial ideals generated by ten forms that fail the WLP in degree 2 are the ideals I = (x 3, y 3, z 3, t 3, f 1, f 2, f 3, f 4, f 5, f 6 ) where the forms f i are chosen among one of the following sets of monomials: (1) {x 2 y, xy 2, x 2 z, x 2 t, y 2 z, y 2 t, z 2 t, zt 2, xyz, xyt}. (Case (A1)) (2) {x 2 y, xy 2, xz 2, y 2 z, yz 2, y 2 t, zt 2, z 2 t}. (Case (A2)) (3) {x 2 y, xy 2, z 2 t, zt 2, xyz, xyt, xzt, yzt}. (Case (A2)) (4) {x 2 y, xy 2, x 2 z, xz 2, x 2 t, xt 2, xyz, xzt, xyt, yzt}. (Case (B1)) Proof. We look for a vector subspace R 3 of dimension 10 that fails the WLP. According to theorem 3.1, we look for a surface of degree 3 with multiplicity 3 at a general point M = (a, b, c, d) (i.e. three concurrent planes) that passes through the points (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1) such that its equation f depends only on the monomials in R 3 /. Remark. One has to write this equation with a number of monomials as small as possible. Then the orthogonal space becomes bigger and we will cover all the possible choices.

10 LAPLACE EQUATIONS AND THE STRONG LEFSCHETZ PROPERTY 9 First of all, there is at least one plane passing through (a, b, c, d) and two basis points; without lost of generality we assume that its equation is dz ct = 0. Then two cases occur. (A) There is another plane through the two remaining points, and one free, i.e. f = (bx ay)(dz ct)(αx + βy + γz + δt), with αa + βb + γc + δd = 0. (B) Each remaining plane passes through one basis point, i.e. f = (dz ct)(u 1 x + v 1 y + w 1 z)(u 2 x + v 2 y + w 2 t) with u 1 ba + v 1 b + w 1 c = u 2 a + v 2 b + w 2 d = 0. Case (A) The equation of f is the following: f(x, y, z, t) = αbdx 2 z αbcx 2 t + d(βb αa)xyz + c(αa βb)xyt βady 2 z + βacy 2 t + γbdxz 2 + b(δd γc)xzt γadyz 2 + a(γc δd)yzt δbcxt 2 + acδyt 2. In order to remove the biggest number of coefficients, different cases occur. (A1) The case α = 0 and β = 0 (i.e. one plane through (0, 0, 1, 0) and (0, 0, 0, 1) and one double plane through (1, 0, 0, 0) and (0, 1, 0, 0). It needs six coefficients to write its equation that are xz 2, xzt, yz 2, yzt, xt 2, yt 2. In R 3 it remains 14 monomials that are the four powers x 3, y 3, z 3, t 3 and ten more x 2 y, xy 2, x 2 z, x 2 t, xyz, xyt, y 2 z, y 2 t, z 2 t, zt 2. The ideal generated by the four powers and six forms among these ten fails the WLP in degree 2. (A2) The case β = 0 and γ = 0. The equation of the cubic is f = (bx ay)(dz ct)(cx az). This equation depends on the eight following monomials Figure 2. cubic with a triple point. Case (A2). yt 2, xyt, yzt, xyz, xt 2, x 2 t, xzt, x 2 z. In R 3 it remains 12 monomials that are (x 3, y 3, z 3, t 3, x 2 y, xy 2, xz 2, y 2 z, yz 2, y 2 t, zt 2, z 2 t). The four powers and eight more. The ideal generated by the four powers and six forms among these eight fails the WLP in degree 2. (A3) The case βb = αa and δd = γc. The supplementary plane do not pass through one of the four basis points. There exists at any point (a, b, c, d) a function u(a, b, c, d) such that ab + u(a, b, c, d)cd = 0. The equation of the supplementary plane is bx + ay + udz + uct = 0. Then the equation of the

11 10 ROBERTA DI GENNARO, GIOVANNA ILARDI AND JEAN VALLÈS cubic depends only on (x 2 z, x 2 t, y 2 z, y 2 t, xz 2, yz 2, xt 2, yt 2 ). In R 3 it remains 12 monomials that are (x 3, y 3, z 3, t 3, xyz, xyt, xzt, yzt, x 2 y, xy 2, z 2 t, zt 2 ). The four powers and eight more. The ideal generated by the four powers and six forms among these eight fails the WLP in degree 2. Case (B) The equation of f is f = (dz ct)(u 1 x + v 1 y + w 1 z)(u 2 x + v 2 y + w 2 t). This equation depends on 14 monomials (i.e. all of them except the four powers and x 2 y, xy 2 ). We have to remove at least the coefficients of four of them. If w 1 = 0 or w 2 = 0 it is again the first situation. We cannot choose u i = v i = 0 because the plane has to pass through (a, b, c, d). (B1) If u 1 = u 2 = 0 we remove 7 coefficients. Then f depends on 6 monomials that are (y 2 z, yz 2, z 2 t, y 2 t, yt 2, zt 2 ). In R 3 it remains 14 monomials that are (x 3, y 3, z 3, t 3, x 2 y, xy 2, x 2 z, xz 2, x 2 t, xt 2, xyz, xzt, xyt, yzt). The four powers and ten more. The ideal generated by the four powers and six forms among these ten fails the WLP in degree 2. Figure 3. cubic with a triple point. Case (B1) (B2) If u 1 = v 2 = 0 the cubic consists in three plan with equations dz ct = 0, cy bz = 0 and dx at = 0. By permutation it corresponds to the case β = γ = 0 which is done above (see figure 2). (B3) If u 1 v 2 + u 2 v 1 = du 1 w 2 cu 2 w 1 = cv 2 w 1 dv 1 w 2 = 0. Then f depends on ten monomials that are (x 2 z, xz 2, x 2 t, xt 2, y 2 z, yz 2, y 2 t, yt 2, z 2 t, zt 2 ). In R 3 it remains 10 monomials that are (x 3, y 3, z 3, t 3, x 2 y, xy 2, xyz, xyt, xzt, yzt). The four powers and eight more. The ideal generated by the four powers and six forms among these eight fails the WLP in degree 2. It appears that it is a subcase of A3. The other cases are obtained by permutation. Remark. If we want of dimension r 10 (for instance 8, like in the examples [9]) we need numerous independent cubics with a triple point. Indeed we have 0 q k p k K Or P S 3 Ω 3 P ( 1) R 1 q 3 k p kk 0, and the kernel is non zero if and only if the generic rank of the cokernel is 10 r + 1 (in other terms 10 r + 1 independent cubics with triple points). As an example let us give two linear systems of eight cubic forms that fail the WLP in degree 2. Proposition 3.5. The following monomial ideals

12 LAPLACE EQUATIONS AND THE STRONG LEFSCHETZ PROPERTY 11 (1) I = (x 3, y 3, z 3, t 3, x 2 y, xy 2, zt 2, z 2 t) and (2) J = (x 3, y 3, z 3, t 3, xyz, xyt, xzt, yzt) fail the WLP in degree 2. Remark. It should be possible to give a complete list by this method. Anyway the complete classification is already done, see [9], Thm Proof. Let us consider the following three forms defining singular cubics passing through the basis points and the general point (a, b, c, d): (ct dz)(at dx)(ay bx) = 0, (ct dz) 2 (ay bx) = 0, (ct dz)(ay bx) 2 = 0. They are specialisations of the case (1) in the proof of theorem 3.4. They are linearly independent and can be written with twelve monomials. Then it remains only 8 forms for I : I = (x 3, y 3, z 3, t 3, x 2 y, xy 2, zt 2, z 2 t). Let us consider the following three forms defining singular cubics passing through the basis points and the general point (a, b, c, d): (bz cy)(az cx)(ay bx) = 0, (bx ay)(at dx)(dy bt) = 0, (az cx)(dx at)(dz ct) = 0. They are specialisations of the case (2) in the proof of theorem 3.4. They are linearly independent and can be written with twelve monomials: It remains only (x 2 y, x 2 z, xy 2, xz 2, y 2 z, yz 2, t 2 y, t 2 z, ty 2, tz 2, t 2 x, x 2 t). J = (x 3, y 3, z 3, t 3, xyz, xyt, xzt, yzt). The ideals I and J correspond respectively to the cases (3) and (1) of the theorem 4.11 in [9]. Case n = 4, d = 3, k = 1. Again we do not give a complete classification but only one example of monomial ideal in k[x, y, z, t, w] that fails the WLP. Proposition 3.6. The following monomial ideal I = (x 3, y 3, z 3, t 3, w 3, xyz, xyt, xyw, xzt, xzw, xtw, yzt, yzw, ytw, ztw), fails the WLP in degree 2. Proof. A cubic solid with a triple point at the general point x = (a, b, c, d, e) is given by three P 3 through x and through the points (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1). Still copying the Togliatti s case we consider the P 3 that contain the P 2 defined by the points (0, 0, 0, 1, 0), (0, 0, 0, 0, 1) and the point (a, b, c, d, e). More precisely we consider three P 3 obtained by choosing a supplementary point among (1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0). Like in the following figure. We have to permute in order to have six independent cubics with triple point. Then it remains at the end I = (x 3, y 3, z 3, t 3, w 3, xyz, xyt, xyw, xzt, xzw, xtw, yzt, yzw, ytw, ztw), and I fails the WLP in degree 2. Case n = 3, d = 4, k = 1. Again we do not give a complete classification but only one example of monomial ideal of degree 4 in k[x, y, z, t] that fails the WLP..

13 12 ROBERTA DI GENNARO, GIOVANNA ILARDI AND JEAN VALLÈS Figure 4. cubic with a triple point. Proposition 3.7. Let f 1,, f 11 be eleven monomials chosen among x 3 y, x 3 z, x 3 t, xy 3, xz 3, xt 3, y 3 z, y 3 t, yz 3, yt 3, z 3 t, zt 3, x 2 y 2, z 2 t 2, y 2 z 2, x 2 t 2. Then the ideal I = (x 4, y 4, z 4, t 4, f 1,, f 11 ) fails the WLP in degree 3. Proof. We want at any point x = (a, b, c, d) a surface of degree 4 with multiplicity 4 at x and that passes through the basis points (1, 0, 0, 0), (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1). It means four concurrent planes. Let us see an example. f = (ct dz)(at dx)(ay bx)(bz cy) = 0. Then we need the following monomials to write this f: yzt 2, xyzt, yz 2 t, xyz 2, xzt 2, x 2 zt, xz 2 t, x 2 z 2, y 2 t 2, xy 2 t, y 2 zt, xy 2 z, xyt 2, x 2 yt, x 2 yz. Twenty monomials are missing. More precisely: x 4, y 4, z 4, t 4, x 3 y, x 3 z, x 3 t, xy 3, xz 3, xt 3, y 3 z, y 3 t, yz 3, yt 3, z 3 t, zt 3, x 2 y 2, z 2 t 2, y 2 z 2, x 2 t 2. Then if we choose 11 monomials f 1,, f 11 among x 3 y, x 3 z, x 3 t, xy 3, xz 3, xt 3, y 3 z, y 3 t, yz 3, yt 3, z 3 t, zt 3, x 2 y 2, z 2 t 2, y 2 z 2, x 2 t 2, the ideal I = (x 4, y 4, z 4, t 4, f 1,, f 11 ) fails the WLP in degree 3. Let us give now an example when k = 2. We look for singular hypersurfaces of degree d with multiplicity d 1 at a point p that consist of d 1 hyperplanes meeting at p and an extra hyperplane such that their union passes through the n + 1 basis points. As before, such a hypersurface has an equation in the canonical monomial basis of degree d forms; if this equation depends only on the monomials of R d / I then the image of v d (P n ) by π : P(R d ) \ P( I ) P(R d / I ) verifies a Laplace equation. Let us give examples. Case n = 2, d = 4, k = 2. Proposition 3.8. The ideal I = (x 4, y 4, z 4, xy 3, xz 3, x 2 yz, y 2 z 2, y 3 z, yz 3 ) k[x, y, z] fails the SLP at the range 2 in degree 2. Proof. We consider the curve of degree 3 through p 1 = (1, 0, 0), p 2 = (0, 1, 0) and p 3 = ((0, 0, 1) with multiplicity 3 in (a, b, c) and the line through p 3 and p 2 i.e. x = 0. Then the union of both curves is a quartic passing through p 1, p 2, p 3 and triple

14 LAPLACE EQUATIONS AND THE STRONG LEFSCHETZ PROPERTY 13 Figure 5. quartique with a quartuple point. Figure 6. quartic with a triple point. at (a, b, c). It depends on the six monomials x 3 y, x 3 z, x 2 y 2, xy 2 z, x 2 z 2, xyz 2. Then it remains 9 = 15 6 monomials I =< x 4, y 4, z 4, xy 3, xz 3, x 2 yz, y 2 z 2, y 3 z, yz 3 > such that the kernel K of the following map 0 K OP 9 I 2 OP 2(4) 0, verifies h 0 (K O L 2) 0 for a general linear form L. According to theorem 3.1 it proves that I = (x 4, y 4, z 4, xy 3, xz 3, x 2 yz, y 2 z 2, y 3 z, yz 3 ) fails the SLP at the range 2 in degree SLP at the range 2 and derivation bundle on P 2 Let us point out the interesting case = H 0 (I Z (n)) where Z is a finite set of points in P 2. This case leads to non toric examples. But mostly, it leads to derivation bundles associated to line arrangements. More precisely, let us denote by f = 0 the equation of

15 14 ROBERTA DI GENNARO, GIOVANNA ILARDI AND JEAN VALLÈS the union of dual lines of the d points of Z and by D 0 (Z) the vector bundle appearing as the kernel of the jacobian map: 0 D 0 (Z) O 3 P 2 ( f) O P 2(d 1). This bundle is called derivation bundle of the line arrangement and it is studied by many different authors (see for instance [14] and [13]). The arrangement of lines is said free with exponents (a, b) when its derivation bundle splits on P 2 as a sum of two line bundles, more precisely when D 0 (Z) = O P 2( a) O P 2( b). The splitting of D 0 (Z) over a line l P 2 is related to the existence of curves (with a given degree d) passing through Z that are multiple (with multiplicity d 1) at l P 2. More precisely, Theorem 4.1. Let Z P 2 be a set of 2n + r + 1 distinct points with n 1, r 0 and l be a general line in P 2. Then the following conditions are equivalent: (1) D 0 (Z) O l = O l ( n) O l ( n r). (2) H 0 ((J Z Jl n )(n + 1)) 0 and H 0 ((J Z J n 1 l )(n)) = 0. Proof. Let us introduce the variety F P 2 P 2. which is the incidence variety point-line in P 2, and the projections p and q on P 2 and P 2. p F q P 2 P 2 We recall first that the derivation bundle is obtained by looking at I Z (1) on P 2. More precisely, we have D 0 (Z) = p q I Z (1) (see [4], prop.1.3). Let us denote by P the blowing up of P 2 along the point x = l. We recall that P p 1 (x ) F and we consider the induced incidence diagram: P p l = x q P 2 Moreover we have the following resolution of P in F: 0 p O P 2( 1) O F O P 0. Then tensoring the exact sequence above by q J Z (1) and taking the direct image by p we obtain: 0 D 0 (Z)( 1) x D 0 (Z) p q J Z (1) [R 1 p q J Z (1)]( 1) x [R 1 p q J Z (1)] R 1 p q J Z (1) 0. Since x is general, any line through x is at most 1-secant to Z. Then the support of the sheaf R 1 p q J Z (1), which is the locus of 3-secant lines to Z through x (by base change theorem, see [11] for instance), is empty. So it proves that we have in fact a short exact sequence: 0 D 0 (Z)( 1) x D 0 (Z) p q J Z (1) 0, and consequently p q J Z (1) = T Z O x. Then the decomposition D 0 (Z) O x = O x ( n) O x ( n r) gives an injective homomorphism: O x ( n) p q J Z (1). This means that we have a non zero map p O x ( n) q J Z (1) on P, that we can write also as: O P q J Z (1) p O x (n).

16 LAPLACE EQUATIONS AND THE STRONG LEFSCHETZ PROPERTY 15 By the projection formula this last map is equivalent to a non zero map on P 2 : O P 2 J Z (1) J n x (n) = (J Z J n x )(n + 1). Let us now prove that H 0 (J Z Jx n 1 )(n)) = 0. Assume that H 0 (J Z Jx n 1 )(n)) 0 and consider a non zero section: O P 2 J Z Jx n 1 (n). The corresponding section on P O x q J Z (n) vanishes in codimension 1 along (n 1)-times the exceptionnal divisor q 1 (x). After simplification by its equation (f H 0 ( p O x (n 1) q O P 2 (1 n))) we obtain: p O x (1 n) q J Z (1). It induces on x a non zero map: O x (1 n) p q J Z (1), contradicting the decomposition of D 0 (Z) along x. In our context it implies the following characterization of unstability. We recall that a rank two vector bundle E on P n, n 2 is unstable if and only if its splitting E l = O l (a) O l (b) on a general line l verifies a b 2. This characterization is a consequence of the Grauert-Mülich theorem, see [11]. Theorem 4.2. Let I R = C[x, y, z] be an artinian ideal generated by 2d + 1 polynomials l d 1,, ld 2d+1 where l i are distinct linear forms in P 2. Let Z = {l 1,, l 2d+1 } be the corresponding set of points in P 2. Then the following conditions are equivalent: (1) The ideal I fails the SLP at the range 2 in degree d 2. (2) The derivation bundle D 0 (Z) is unstable. Proof. The failure of SLP at the range 2 in degree d 2 is equivalent to the existence at a general point l of a curve of degree d with multiplicity d 1 at l belonging to = H 0 (I Z (d)). But, the description of D 0 (Z) as the image of I Z (1) implies that such a curve imposes the splitting of D 0 (Z) over the line l. More precisely we have the following equivalence: O l (1 d) D 0 (Z) O l H 0 ((I Z m d 1 l )(d)) 0. Then on the general line D 0 (Z) O l = O l (d s) O l (d + s) with s > 0. In other words it gives a non balanced decomposition and according to Grauert-Mülich theorem it is equivalent to unstability, see [11]. Let us give now an ideal (not generated by monomials) that fails the SLP at the range 2. It comes from a line arrangement such that the derivation bundle associated is unstable (in fact even decomposed). Example 4.3. Consider the set of nine points Z = {(1, 0, 0), (0, 1, 0), (0, 0, 1), (1, 1, 0), (1, 1, 0), (1, 0, 1), (1, 0, 1), (0, 1, 1), (0, 1, 1)}. The dual linear forms are x, y, z, x + y, x y, x + z, x z, y + z, y z. We consider the map 0 K OP 9 φ O 2 P 2(4) 0, where φ(a 1,, a 9 ) = a 1 x 4 +a 2 y 4 +a 3 z 4 +a 4 (x+y) 4 +a 5 (x y) 4 +a 6 (x+z) 4 +a 7 (x z) 4 +a 8 (y+z) 4 +a 9 (y z) 4. The associated derivation bundle is D 0 (Z) = O P 2( 3) O P 2( 5) (it is free with exponents (3, 5); see [12] for a proof). Then, according to theorem 4.1 there is at any point p a degree 4 curve with multiplicity 3 at p passing through Z. In other words we have H 0 (K O L 2) 0 for a general linear form L and we have proved that:

17 16 ROBERTA DI GENNARO, GIOVANNA ILARDI AND JEAN VALLÈS Proposition 4.4. The ideal I = (x 4, y 4, z 4, (x + y) 4, (x y) 4, (x + z) 4, (x z) 4, (y + z) 4, (y z) 4 )k[x, y, z] fails the SLP at the range 2 and degree 2. More generally we can produce examples that come from free arrangements. Proposition 4.5. Let A = {l 1,, l a+b+1 } a line arrangement that is free with exponents (a, b) such that a b, b a 2 and a + b even. The ideal I = (l a+b 2 1,, l a+b 2 a+b+1 ) fails the SLP at the range 2 and degree a+b 1 2. Remark. If a + b is odd we can add to Z one point p in general position with Z and we can prove in the same way that I = (l a+b+1 2 1,, l a+b+1 2 a+b+1, (p ) a+b+1 2 ) fails the SLP at the range 2 and degree a+b 2. Proof. Let us denote by Z = {l1,, l a+b+1 } the dual set of points of A. Since there exists at any general point l a curve of degree a + 1 passing through Z theorem 4.1 implies that D 0 (Z) is unstable and theorem 4.2 implies that I fails the SLP at the range 2 and degree a+b SLP at the range 2 and Terao s conjecture. One of the main conjecture about hyperplane arrangements (still open also for line arrangements) is the Terao s conjecture. It concerns the arrangements that are called free. The conjecture says that freeness depends only on the combinatorics of the arrangement (the combinatorics of the arrangement A = {l 1,..., l n } is encoded by a incidence graph with vertices given by the lines l i and the points l i l j and edges joining point and line when the point belongs to the line). We refer to [12] for a good introduction to the subject. Conjecture (Terao). The freeness depends only on combinatorics. In other words if an arrangement has the same combinatorics than a free arrangement it is also free. Let us consider an arrangement of lines and let us denote by Z 0 its dual set of points. We assume that Z 0 is free with exponents (a, b) with a b. We assume that there exists a non free arrangement Z with the same combinatorics than Z 0, i.e. that the Terao s conjecture is not true. Let us add b a points in general position to Z 0 in order to form Γ 0 and to Z to form Γ. Then the length of both sets of points is 2b + 1. On the general line l we have D 0 (Z 0 ) O l = O l ( a) O l ( b), when, since Z is not free we have a less balanced decomposition for D 0 (Z) (this affirmation is not so easy; it is proved in [2]), more precisely It implies that D 0 (Z) O l = O l (s a) O l ( s b), s 1. H 0 (I Z m a 1 l (a)) 0, H 0 (I Z0 m a 1 l (a)) = 0 and H 0 (I Z0 m a l (a + 1)) 0. Then adding b a lines passing through l and the b a added points we obtain H 0 (I Γ m b 1 l (b)) 0, H 0 (I Z0 m b 1 l (b)) = 0 and H 0 (I Z0 m b l (b + 1)) 0, in other words D 0 (Γ 0 ) is balanced with splitting O l ( b) O l ( b) when D 0 (Γ) O l = O l (1 b) O l ( 1 b).

18 LAPLACE EQUATIONS AND THE STRONG LEFSCHETZ PROPERTY 17 Then D 0 (Γ) is unstable. In other words if Z 0 = {d 1,, d a+b+1 }, Γ 0 = {d 1,, d a+b+1, p 1,, p b a }, Z = {l 1,, l a+b+1 } and Γ = {l 1,, l a+b+1, p 1,, p b a } then the ideal (l b 1,, l b a+b+1, (p 1 ) b,, (p b a )b ) fails the SLP at the range 2 and degree b 2 when (d b 1,, d b a+b+1, (p 1 ) b,, (p b a )b ) has SLP at the range 2 and degree b 2. The following conjecture written in terms of SLP implies the Terao s conjecture on P 2. Conjecture. Let Z 0 = {d 1,, d 2b+1 } a set of points of length 2b+1 in P2 such that the ideal I = (d b 1,, db 2b+1 ) has the SLP at the range 2 and degree b 2. Assume that Z = {l1,, l 2b+1 } has the same combinatorics than Z 0. Then J = (l1 b,, lb 2b+1 ) has the SLP at the range 2 and degree b 2. References [1] Holger Brenner and Almar Kaid. Syzygy bundles on P 2 and the Weak Lefschetz Property. Illinois J. Math., 51: , [2] Georges Elencwajg and Otto Forster. Bounding cohomology groups of vector bundles on P n, Math. Ann., 246(3): , 1979/80. [3] Davide Franco and Giovanna Ilardi. On a theorem of Togliatti. Int. Math. J. 2(4): , [4] Daniele Faenzi, Daniel Matei and Jean Vallès. Hyperplane arrangements of Torelli type. ArXiv e-prints, November To appear in Compositio Math. [5] Joe Harris. Algebraic geometry, a first course. volume 133 of Graduate Texts in Math. Springer Verlag, [6] Giovanna Ilardi. Rational varieties satisfying one or more Laplace equations. Ricerche di Matematica, 48(1): , [7] Giovanna Ilardi. Togliatti systems. Osaka J. Math., 43(1):1 12, [8] Antonion Lanteri and Raquel Mallavibarena. Osculatory behavior and second dual varieties of Del Pezzo surfaces. Adv. in Geom., 2(4): , [9] Emilia Mezzetti, Rosa Miró-Roig and Giorgio Ottaviani G. Laplace Equations and the Weak Lefschetz Property, ArXiv e-prints, October To appear in Canadian Journal of Mathematics. [10] Juan C. Migliore and Uwe Nagel. A tour of the weak and strong Lefschetz properties. ArXiv e-prints, September [11] Christian Okonek, Michael Schneider and Heinz Spindler, Vector bundles on complex projective spaces. Progress in Mathematics, vol. 3, Birkhäuser, Boston, Mass., [12] Peter Orlik and Hiroaki Terao, Arrangement of hyperplanes, volume 300 of Grundlerhen der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, [13] Kyoji Saito, Theory of logarithmic differential forms and logarithmic vector fields. J. Fac. Sci. Univ. Tokyo Sect. IA Math., 27(2): , [14] Henry K. Schenk. Elementary modifications and line configurations in P 2. Comment. Math. Helv. 78(3): , [15] A. Terracini. Sulle V k che rappresentano più di 1 k(k 1) equazioni di Laplace linearmente 2 indipendenti. Rend. Circ. Mat. Palermo, 33: , [16] Eugenio Togliatti. Alcune osservazioni sulle superficie razionali che rappresentano equazioni di Laplace. Ann. Mat. Pura Appl. 25(4): , [17] Jean Vallès. Variétés de type Togliatti. C.R.A.S. 343(6): , 2006.